Canonical solution of the SU(3) ↓ SO(3) reduction problem from the SU(3) pattern calculus

Abstract

Because of its numerous applications to physics, there have been many solutions published on the problem of reducing a given irreducible representation (irrep) of the unitary unimodular group SU(3) into irreps of the proper orthogonal subgroup SO(3). Such solutions are generally based on an arbitrary construction of a nonorthogonal basis of the highest weight space for an irrep of SO(3), followed by an equally arbitrary orthonormalization procedure. This paper presents a unique solution of this problem based on the intrinsic structure of the multiplicity function, which is a function M L(p, q) giving the number of times irrep[L] of SO(3) is contained in irrep[pq0] of SU(3). This structure is implemented uniquely into the reduction problem through the use of the SU(3) pattern calculus.